Lecture.08-Homework_Solutions

# Lecture.08-Homework_Solutions - ECE155A Fall 07...

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ECE155A Fall 07 Equalization Homework Solutions Part I (Homework 1) 1. Note : This problem is not very clear stated. Therefore the points won’t be counted. The question is restated as following: Find Z-transform and DTFT of a n+m u ( n ) , | a | < 1, u ( n ) is a step function , u ( n ) =0 for n<0 and u ( n )=1 for n>1. ‘m’ is an arbitrary constant. Solution: The Z-transform of a n u ( n ) is 1/(1-az -1 ). Since a n+ m u ( n ) = a m ( a n u ( n )), its Z-transform is a m /(1- a z -1 ). Substituting z=e jw , have the DTFT transform as a m /(1- a e -jw ) 2. (10pt) Solution: Let S be the sum we seek. (Note: we assume that the sum converges without proof). Then S=1+q+q 2 +q 3 +…= 1 + q(1+q+q 2 +q 3 +…) = 1+qS. Solving this equation for S, have S=1/(1-q). 3. (10pt) Solution: Let y(n) be the output of the channel. −∞ = = + = = k n qx n x n x h n x h k n x k h n y ) 1 ( ) ( ) 1 ( ) 1 ( ) ( ) 0 ( ) ( ) ( ) ( n Input x(n) Response y(n) 0 0 0 1 1 1 q q-q=0 2 q 2 q 2 -qq=0 3 q 3 q 3 -qq 2 =0 4 q 4 q 4 -qq 3 =0 0 Hence, the answer is that the output y(n)=1, n=0 , and y(n)=0 for n 0 4. (10pt) Solution : Let y(n) be the output of the channel. ) 1

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## This note was uploaded on 11/11/2009 for the course ECE 670377 taught by Professor Wolf during the Winter '07 term at UCSD.

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Lecture.08-Homework_Solutions - ECE155A Fall 07...

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